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In the present paper we discuss the boundedness of the maximal operator in the Lorentz space of variable exponent defined by the symmetric decreasing rearrangement in the sense of Almut [1]. As an application of the boundedness of the maximal operator, we establish the Sobolev inequality by using Hedberg’s trick in his paper [10].

Sobolev's inequality for Riesz potentials in Lorentz spaces of variable exponent